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Volatility of a stock may incur a risk premium, leading to a positive correlation between volatility and returns. On the other hand the leverage effect, whereby negative returns increase volatility, acts in the opposite direction. We propose a reformulation and extension of the ARCH in Mean model, in which the logarithm of scale is driven by the score of the conditional distribution. This EGARCH-M model is shown to be theoretically tractable as well as practically useful. By employing a two component extension we are able to distinguish between the long and short run effects of returns on volatility. The EGARCH formulation allows more flexibility in the asymmetry of the response (leverage) and this enables us to find that the short-term response is, in some cases, close to being anti-asymmetric. The long and short run volatility components are shown to have very different effects on returns, with the long-run component yielding the risk premium. A model in which the returns have a skewed t distribution is shown to fit well to daily and weekly data on some of the major stock market indices.

Beta-t-EGARCH models in which the dynamics of the logarithm of scale are driven by the conditional score are known to exhibit attractive theoretical properties for the t-distribution and general error distribution (GED). The generalized-t includes both as special cases. We derive the information matrix for the generalized-t and show that, when parameterized with the inverse of the tail index, it remains positive de nite as the tail index goes to infi nity and the distribution becomes a GED. Hence it is possible to construct Lagrange multiplier tests of the null hypothesis of light tails against the alternative of fat tails. Analytic expressions may be obtained for the unconditional moments in the EGARCH model and the information matrix for the dynamic parameters obtained. The distribution may be extended by allowing for skewness and asymmetry in the shape parameters
and the asymptotic theory for the associated EGARCH models may be correspondingly extended. For positive variables, the GB2 distribution may be parameterized so that it goes to the generalised gamma in the limit as the tail index goes to in finity. Again dynamic volatility may be introduced and properties of the model obtained. Overall the approach offers a uni ed, flexible, robust and practical treatment of dynamic scale.

The chapter develops a simple model for the optimal path of carbon emissions over the next century. It contrasts two extreme cases: where the cost of reducing carbon emissions is purely “pay-as-you go” (i.e. no long-term effects), and where the cost is purely a transitional investment (i.e. only the transition to a greener energy system is costly). In both cases, we solve a variational problem to show the optimal path of carbon emissions over the next 100 years. We show that it makes economic sense to invest, in each time period, a multiple of contemporaneous climate change damages.

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schr\”odinger’s equation. This paper, in contrast, investigates the integral form of Schr\”odinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schr\”odinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schr\”odinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s result to hypersurfaces. Second, we derive a new closed-form solution to Schr\”odinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schr\”odinger’s differential equation. Third, we derive boundary conditions for `super-singular’ potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schr\”odinger’s integral equation is viable tool for studying singular interactions in quantum mechanics.

This paper postulates a distribution function as the potential in the heat/Schrodinger equation: the Laplacian of the indicator function of some domain in d dimensions. This distributional function can be viewed as a surface delta prime function. To see why this is the case, compare with the second derivative of the Heaviside step function in one dimension. The paper derives that the resulting solution satisfies Dirichlet or Neumann boundary conditions, depending on the sign of the potential, thereby establishing for the first time a particularly close connection between these two classical boundary value problems. When we postulate in addition a surface delta function, we can also replicate the solution to the Robin boundary value problem. The paper thus establishes an explicit link between a new set of singular potentials and a set of classical boundary conditions.

This paper is a shortened (in length) and extended (in content) version of Part I of my PhD thesis and appeared in the Journal of High Energy Physics in November 2012. A working paper version of the paper can be found on this website (here), or on the ArXiv (here). The published paper can be found here.